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LDC-2011-0013/9/2011
A LINEARLY IMPLICIT ALGORITHM FOR INTEGRATING
THE EQUATIONS OF CHEMICAL KINETICS
Lawrence D. Cloutman
Abstract
The rate equations of chemical kinetics are a system of nonlinear ordinary differ-ential equations. Explicit numerical integration methods tend to have restrictive timestep limitations, so there is an incentive to use implicit methods with better stabilityproperties. However, the non-linear character of these equations makes numerical so-lution challenging. We present a method of linearizing the difference equations in theadvanced-time terms that will provide improved numerical performance over explicitmethods while remaining numerically tractable. The applications include regular chem-ical reaction networks, nuclear chemistry networks, and the Lotka-Volterra equationsof theoretical ecology.
c2011 by Lawrence D. Cloutman. All rights reserved.
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1 Introduction
The equations of chemical kinetics are a system of nonlinear ordinary differential equations.
In many combustion applications, these rate equations are incorporated into the multicompo-
nent partial differential equations of fluid dynamics. In that case, the rate equations typically
are operator-split from the fluid equations and solved as a system of ordinary differentialequations.
Many methods have been used to solve the rate equations. Explicit integration meth-
ods tend to have restrictive time step limitations, so there is an incentive to use implicit
methods with better stability properties. However, the non-linear character of these equa-
tions makes numerical solution challenging. We present a method of linearizing the difference
equations in the advanced-time terms that will provide improved numerical performance over
explicit methods while remaining numerically tractable. The method described here is a spe-
cialization of a commonly used approach described in Appendix A [1]. This report documents
its implementation into an updated version of an existing program [2].
The proposed method is not limited to combustion research. It is applicable to any
system of coupled nonlinear ordinary differential equations. Other applications include non-
combustion chemical reaction networks, nuclear chemistry networks (including astrophysical
applications), and the Lotka-Volterra equations of theoretical ecology [3].
In Section 2, we describe the governing equations and the constitutive relations used
in combustion research. Section 3 details the rate expressions for chemical reaction networks.
Section 4 defines and describes some of the simple and commonly used methods for solving
the rate equations, plus a discussion of some of their numerical properties. Section 5 describesthe linearized implicit algorithm for integrating the rate equations. Sections 6 and 7 present
some numerical examples, plus the analytic solution for a single-reaction example. Section
8 contains the summary and conclusions. There are two appendixes providing additional
details on selected topics. All units are CGS and the temperature T is in K unless otherwise
noted.
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2 Governing Equations
To solve reactive flow problems, we assume the fluid is a mixture of species described by
the single-velocity (mass weighted) representation. The equations outlined in this section
have been incorporated into an updated version of the COYOTE computer program [2] to
produce the numerical results presented here.Mass conservation is expressed by the continuity equation for each species :
t
+ (u) = J + R, (1)
where is the density of species , t is time, u is the velocity, and R is the rate at which
species is created by chemical reactions. The exact diffusional mass flux is given by an
extremely complicated expression [4, 5, 6]. A clear summary of the mass transport equations
and an efficient numerical algorithm for solving them are given by Ramshaw [7, 8] and by
Ramshaw and Chang [9]. Equation (1) may be summed over species to obtain the total
continuity equation
t+ (u) = 0. (2)
The momentum equation is
(u)
t+ (uu) = g P + T, (3)
where g is the gravitational acceleration, P is the pressure, and T is the stress tensor
T =
u + (u)T 2
3 u U
+ b u U, (4)
where U is the unit tensor, is the coefficient of viscosity, and b is the bulk viscosity.
We can express energy conservation in terms of the specific thermal internal energy
I:(I)
t+ (Iu) = P u + T : u q +
HR, (5)
where q is the diffusional heat flux, and H is the heat of formation of species (per unit
mass). The heat flux is a complicated function, and for many applications it is adequate to
use the sum of Fouriers law and enthalpy diffusion:
q = KT +
h
J
, (6)
where K is the thermal conductivity.
In most cases, we prefer to use the total energy density E = I+ 0.5u u + , where
is the gravitational potential.
E
t+ [ (E + P)u u T] =
t q +
HR. (7)
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For a constant g, = g x, where x is the position vector.
The thermal equation of state is the sum of partial pressures for each species:
P =
RT
M, (8)
where R is the universal gas constant, and M is the molecular weight of species .The caloric equation of state is
I =
I(T), (9)
where I is the species specific thermal internal energy. In the present application, we assume
I = CvT =RT
( 1)M, (10)
where is the ratio of specific heats for species .
These same equations for chemical combustion apply to nucleosynthesis in stellar
interiors, although the stress tensor and constitutive relations may need to be modified to
include additional physical processes such as radiative effects and electron degeneracy [10,
11].
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3 Chemical Reaction Networks
Chemical reactions are symbolized by
ar
br, (11)
where represents one mole of species , and ar and br are the dimensionless stoichio-
metric coefficients for the rth reaction. It is assumed that ar and br are integers. The
chemical source term in the species continuity equation is given by
R = Mr
(br ar)r, (12)
where r is the rate of progress of the rth reaction:
r =
kfr
M
ar kbr
M
brCrM moles(of reaction)/cm
3s. (13)
Here kfr and kbr are the forward and reverse rate coefficients for reaction r. For an elementary
reaction, ar = ar and b
r = br. The product operations are over all species for which
the stoichiometric coefficients ar and br are nonzero. For combustion, the coefficients kfr
and kbr are assumed to be of a generalized Arrhenius form:
kfr = AfrTfr exp(Efr/RT), (14)
kbr = AbrTbr exp(Ebr/RT), (15)
where Efr and Ebr are the activation energies. The expressions for nuclear reaction rates are
similar but tend to be a bit more complex [12].
In most cases, the constants Abr, br, and Ebr are computed from Afr, fr, Efr, and
the reactions equilibrium constant KCr . In equilibrium, r = 0, so
kbrkfr
=
M
ar| /
M
br= KCr . (16)
The factor CrM requires some explanation. Some chemical combustion reactions in-
clude the modified total molar density
CM =N=1
n
M, (17)
where is the third body efficiency (or chaperon coefficient) for species (normally unity
unless otherwise specified). An example of such a reaction is 2 H + M H2 + M, where M
represents all of the species in the mixture. The parameter r is zero or unity depending on
whether M occurs in the reaction.
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4 Some Commonly-Used Numerical Methods
The chemical reaction terms are usually time-split from the rest of the fluid dynamics equa-
tions so the numerical chemistry problem reduces to a system of ordinary differential equa-
tions in each computational zone on each time step. We shall consider several simple methods
for numerically integrating the system of ordinary differential equations
dx
dt= f(x, t) (18)
for the solution vector x.
The simplest algorithm is the first-order Eulers method, the fully explicit method for
advancing from time ti1 to ti,
xi = xi1 + t f(xi1, ti1), (19)
where t = ti ti1 and the subscript i denotes the value of the function at time ti.Next consider the fully implicit first-order method
xi = xi1 + t f(xi, ti). (20)
In general, we must solve nonlinear algebraic or transcendental equations for xi. A solution
algorithm is outlined in Appendix A.
Now we shall consider some second-order methods. First, the frequently-used Crank-
Nicholson scheme is
xi = xi1 +t
2 [f(xi1, ti1) + f(xi, ti)] . (21)
Next we consider a two-step second-order Runge-Kutta scheme based on the midpoint
rule.
xi = xi1 +t
2f(xi1, ti1)
xi = xi1 + t f(x, ti1/2), (22)
where ti1/2 = 0.5(ti + ti1).
Finally we consider a second-order Runge-Kutta scheme based on the trapezoidal
rule.
xi = xi1 + t f(xi1, ti1)
xi = xi1 +t
2[f(xi1, ti1) + f(x
, ti)] . (23)
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For the limit of vanishing t to make sense, we must choose the minus sign regardless of
the sign of K. For K < 0, the solution decays and stays positive for all values of t, as
it should. However, for K > 0 and sufficiently large t, the argument of the square root
becomes negative.
Now we shall consider some second-order methods. First, the Crank-Nicholson scheme,
equation (21), is
xi = xi1 +K t
2
xmi1 + x
mi
. (31)
For m = 1,
xi =xi1 (1 + 0.5 K t)
1 0.5 K t. (32)
Regardless of the sign ofK, the solution will alternate its sign from one time step to the next
if t exceeds twice the explicit limit. The same thing happens for the diffusion equation in
spite of a formal stability analysis showing that the method is unconditionally stable. That
is, the solution remains bounded, but it is bounded garbage. For m = 2, the Crank-Nicholsonmethod has qualitatively the same issues as the fully implicit method.
Next we consider the two-step Runge-Kutta scheme based on the midpoint rule,
equation (22).
xi = xi1 +K t
2xmi1
xi = xi1 + K t (x
i )m . (33)
These may be combined algebraically to obtain
xi = xi
1
1 + 2 + 2
2(m = 1) (34)
and
xi = xi11 + 2xi1 + 4(xi1)
2 + 2(xi1)3
(m = 2), (35)
where = 0.5 K t. We note the following behaviors:
For K > 0, both solutions solution grow along with t, just as they should.
For K < 0 and m = 1, the method is only conditionally stable (in the sense of
maintaining xi 0). The polynomial in has a minimum at = 0.5 (that is,
t = 1/K). Physically, we expect xi to decrease more on each time step as t is
increased. This fails to happen for < 0.5 (remember that becomes more negative
with increasing t if K < 0).
Things are even worse for m = 2 and K < 0: The polynomial has two extrema (at
= 1/3 and 1) and begins to behave badly for xi1 < 1/3.
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Finally we consider the Runge-Kutta scheme based on the trapezoidal rule, equation (23).
xi = xi1 + K t xmi1
xi = xi1 +K t
2
xmi1 + (x
i )m
. (36)
The combined equations are
xi = xi1
1 + 2 + 22
(m = 1) (37)
and
xi = xi11 + 2xi1 + 4(xi1)
2 + 4(xi1)3
(m = 2). (38)
The behavior is qualitatively the same as for the previous method. For m = 1, the two
Runge-Kutta methods are identical. For m = 2, the polynomial factor goes negative for
xi1 less than about 0.75.
The method of most interest for this report is linear in the advanced time variables.
For the case m > 1, it is obtained by by approximating the fully implicit equation (27) with
xmi mxixm1i1 (m 1)x
mi1. The solution may be written as
xi = xi11 2(m 1) xm1i1
1 2 m xm1i1. (39)
The method fails if mxm1i1 > 0.5. All seems well for negative values of . However, as
xm1i1 approaches , the ratio xi/xi1 saturates at a value of (m 1)/m rather than
zero. This means that while the method may be stable for very large time steps, transientsolutions may be quite inaccurate.
A second method of linearization is xmi xixm1i1 . The solution may be written as
xi =xi1
1 2xm1i1. (40)
While this method has the correct limit for large t, it is not obvious how to apply it to a
term such as xiyi, where x and y are the molar concentrations of two different species.
There are two main lessons from this discussion. First, implicitness is not always the
solution to numerical stability issues. Second, numerical stability does not imply accuracy.Indeed, explicit stability limits often play the additional role of accuracy conditions. We note
that there is another, more subtle failure mode for nonlinear difference equations, which is
described in Appendix B.
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5 Linearized Implicit Chemistry Algorithm
The species continuity equation (1) is solved in COYOTE by operator splitting. That is,
the advection, diffusion, and chemical terms are computed independently. The chemistry
problem involves the reactions
ar
br. (41)
The operator-split chemical rate equations are
(x, t)
t= R[T(x, t), 1(x, t), . . . , N(x, t)]
= Mr
(br ar)
kfr(T)
y(r)=react
Cayry kbr(T)
y(r)=prod
Cbyry
CrM, (42)
where there are N species and 1 N. The algebraic product operations are over
chemical products and reactants in an obvious notation. For elementary reactions, ar = ar
and br = br. The parameter r is zero unless the reaction involves species that do not
change during the reaction (typically denoted by M in reactions such as H2 + M 2 H +
M), in which case it is unity. The energy released per mole of reaction is
Qr =
(ar br)HM, (43)
where H is the heat of formation per unit mass.
The original version of the program used the explicit method
t
=n+1
n
t= R (T
n, n1 , . . . , nN) , (44)
where t is the time step and the superscript n denotes the solution at time tn. Both stability
and (possibly) accuracy require a time step that is smaller than the time scale for the fastest
reaction. If we replace all occurrences of n on the right-hand side of equation (44) with
n + 1, we have a fully implicit method. While this method should be much more stable, it
is more expensive and difficult to solve. We have also tried two second-order Runge-Kutta
methods. While these perform somewhat better than the explicit method, they still have
similar accuracy and stability requirements.We propose a method that is linear in advanced-time quantities. It should be more
stable than the fully explicit method, but simpler to solve than the fully implicit method.
First, we use the temperature at time level n. It is more convenient to do the numerical
computations in terms of the species molar concentrations C = /M, which we take at
time level n + 1. Finally, we linearize the products of concentrations to make them linear
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in time level n + 1 quantities. If we assume Cn+1 = Cn + C, then for a product of Np
species,Npy=1
Cn+1y
my
Npy=1
Cny
my1 + Np
y=1
myCyCny
, (45)
where my are constants, usually positive integers.2 Note that linearization is equivalent to
that used in equation (39).
The rate equations are discretized fully implicitly in the species densities, but explic-
itly in temperature:
1
M
t
=r
(br ar)
kfr(Tn)
y(r)=react
n+1yMy
ayr kbr(T
n)
y(r)=prod
n+1yMy
byr CrM.(46)
Next we use equation (45) to linearize the equations and to produce a linear system in the
. Since the total molar concentration of the mixture will change little during the time
step, we compute CM at time n:
CM =N=1
n
M, (47)
where the are the third body efficiencies (also called the chaperon coefficients). Rewriting
equation (46) in terms of concentrations,
Ct
=r
(br ar)
kfr(Tn)
y(r)=react
Cn+1y
ayr kbr(T
n)
y(r)=prod
Cn+1y
byr CrM. (48)Linearizing by using equation (45), we obtain
C t CrM
r
(br ar)
kfr(Tn)
y(r)=react
Cny
ayry
ayrCyCny
kbr(Tn)
y(r)=prod
Cny
byry
byrCyCny
= t CrM
r(br ar)
kfr(T
n)
y(r)=react Cny
ayr
kbr(Tn)
y(r)=prod Cny
byr
. (49)
This linear system is not singular. The mass conservation condition
=
M C = 0 (50)
2Occasionally one encounters rate expressions that involve non-integer powers. Those cases are generallyin simplified global rate expressions (for example, [14]), and they can be handled using the above linearexpansion in the . However, those will not be considered explicitly here.
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is redundant and need not be included in the linear system. However, should we ever
encounter conditions where the matrix is ill-conditioned, we might want to try replacing one
of the species equations with this one.
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6 Example 1: A Single Elementary Chemical Reaction
Consider the single two-body elementary reaction H2 + O2 2 OH. The rate equations
(r = 1) aredH2
dt= RH2 = MH2 kf1CH2CO2 kb1C
2OH , (51)
dO2dt
= RO2 =MO2MH2
RH2 , (52)
anddOH
dt= ROH = RH2 RO2. (53)
To simplify the notation, let us assign k = 1, 2, and 3 for H2, O2, and OH, respectively.
Then we may rewrite our three rate equations as
dC1dt
= kf1C1C2 kb1C
23
, (54)
dC2dt
= kf1C1C2 kb1C
23
, (55)
anddC3dt
= 2kf1C1C2 kb1C
23
. (56)
6.1 Analytic Solution
We begin solving equations (54) through (56) by finding invariants of the problem. First,
adding the equations shows that the total molar density
C = C1(t) + C2(t) + C3(t) = C1(0) + C2(0) + C3(0) (57)
is constant. 3 Comparing pairs of equations shows that
dC1dt
=dC2dt
= 1
2
dC3dt
. (58)
Integrating these equalities gives us the constraints
C1(t) C1(0) = C2(t) C2(0) = 1
2[C3(t) C3(0)] . (59)
If we solve equation (59) for C1(t) and C2(t) in terms of C3(t), we can eliminate
these functions from equation (56). If we let bi = Ci(0) + 0.5 C3(0), i = 1 or 2, thenCi(t) = bi 0.5 C3(t). Then equation (56) becomes
dC3dt
= 2kf1b1b2 kf1 (b1 + b2) C3 +
kf1
2 2kb1
C23 B0 + B1C3 + B2C
23 . (60)
3Total molar concentration is not always conserved; for example 2 H + M H2 + M changes the numberof moles.
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The constants B0, B1, and B2 may be evaluated from the initial conditions.
Equation (60) may be solved by using the following integrals. If B21 > 4B0B2, then
dx
B2x2 + B1x + B0=
1
(B21 4B0B2)1/2
ln
2B2x + B1 (B21 4B0B2)1/2
2B2x + B1 + (B21 4B0B2)1/2
. (61)
If B21 < 4B0B2, then
dx
B2x2 + B1x + B0=
2
(4B0B2 B21)1/2
arctg
2B2x + B1
(4B0B2 B21)1/2
. (62)
If B21 = 4B0B2, then dx
B2x2 + B1x + B0=
2
2B2x + B1. (63)
The corresponding solutions for equation (60) are easily found. First, let
A =B21 4B0B21/2 . (64)
If B21 > 4B0B2, then let
D =1
Aln
2 B2 C3(0) + B1 A
2 B2 C3(0) + B1 + A
. (65)
Then the solution is
C3(t) =(B1 + A) exp [A (t + D)] B1 + A
2 B2 {1 exp[A (t + D)]}. (66)
If B21 < 4B0B2, let
E =2
Aarc tg
2 B2 C3(0) + B1
A
. (67)
Then the solution is
C3(t) =A tan[A (t + E) /2] B1
2 B2. (68)
If B21 = 4B0B2, then
C3(t) = 12B2
12B2C3(0) + B1
t2
1 B1
. (69)
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8.28624D-01 analytic
Species Mass Fractions, ncyc = 2000 t = 1.000000D-061.56861D-01 9.88203D-03 8.33257D-01 Euler1.56863D-01 9.88213D-03 8.33255D-01 midpoint RK1.56864D-01 9.88223D-03 8.33254D-01 linearized
8.33255D-01 analytic
This problem was solved for t = 0.5 ns using three methods: the first-order explicit Eu-
lers method, the second-order midpoint Runge-Kutta method, and the first-order linearized
implicit method. The analytic solution is also shown in Table 1. All three methods agree
with one another and with the analytic solution. To further demonstrate grid-independence
of the solution, the calculation was repeated with the time step cut in half. The results are
shown in Table 2.
+++++ Table 2. Cut delt in half (0.25 ns): jchem2.0.5 +++++
Species Mass Fractions, ncyc = 1 t = 2.500000D-109.35654D-01 5.89449D-02 5.40144D-03 Euler9.35678D-01 5.89464D-02 5.37537D-03 midpoint RK9.35702D-01 5.89479D-02 5.34978D-03 linearized
Species Mass Fractions, ncyc = 2 t = 5.000000D-109.30898D-01 5.86453D-02 1.04569D-02 Euler9.30946D-01 5.86483D-02 1.04054D-02 midpoint RK9.30994D-01 5.86513D-02 1.03549D-02 linearized
1.04056D-02 analytic
Species Mass Fractions, ncyc = 40 t = 1.000000D-087.80330D-01 4.91597D-02 1.70510D-01 Euler7.80954D-01 4.91990D-02 1.69847D-01 midpoint RK7.81568D-01 4.92377D-02 1.69194D-01 linearized
1.69850D-01 analytic
Species Mass Fractions, ncyc = 2000 t = 5.000000D-071.61191D-01 1.01548D-02 8.28655D-01 Euler1.61219D-01 1.01566D-02 8.28624D-01 midpoint RK1.61248D-01 1.01584D-02 8.28594D-01 linearized
8.28624D-01 analytic
Species Mass Fractions, ncyc = 4000 t = 1.000000D-061.56862D-01 9.88208D-03 8.33256D-01 Euler
1.56863D-01 9.88213D-03 8.33255D-01 midpoint RK1.56863D-01 9.88218D-03 8.33254D-01 linearized8.33255D-01 analytic
We see that the agreement is even better. The solution at late times should approach the
equilibrium condition,
KC =kbkf
=C1C2
C23. (74)
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Indeed, the solution is converging toward the correct equilibrium value C3(t) = 0.833333.
Next we try solutions at three larger time steps. The first solution uses t = 5.0 ns
and is shown in Table 3.
+++++ Table 3. Increase delt by 10 (5 ns) +++++
Species Mass Fractions, ncyc = 1 t = 5.000000D-098.44362D-01 5.31936D-02 1.02445D-01 Euler8.53940D-01 5.37971D-02 9.22626D-02 midpoint RK8.60668D-01 5.42209D-02 8.51110D-02 linearized
9.29626D-02 analytic
Species Mass Fractions, ncyc = 100 t = 5.000000D-071.60661D-01 1.01214D-02 8.29217D-01 Euler1.61237D-01 1.01577D-02 8.28605D-01 midpoint RK1.61824D-01 1.01947D-02 8.27981D-01 linearized
8.28624D-01 analytic
Species Mass Fractions, ncyc = 200 t = 1.000000D-06
1.56848D-01 9.88122D-03 8.33271D-01 Euler1.56863D-01 9.88215D-03 8.33255D-01 midpoint RK1.56880D-01 9.88320D-03 8.33237D-01 linearized
8.33255D-01 analytic
Species Mass Fractions, ncyc = 4000 t = 2.000000D-051.56789D-01 9.87750D-03 8.33333D-01 Euler1.56789D-01 9.87750D-03 8.33333D-01 midpoint RK1.56789D-01 9.87750D-03 8.33333D-01 linearized
Although accuracy is slightly degraded, it is still acceptable for most purposes for all three
methods.
+++++ Table 4. Increase delt by 100 (50 ns) +++++
Species Mass Fractions, ncyc = 1 t = 5.000000D-08-2.05086D-02-1.29201D-03 1.02180D+00 Euler
7.20625D-01 4.53984D-02 2.33976D-01 midpoint RK6.24729D-01 3.93570D-02 3.35914D-01 linearized
5.02088D-01 analytic
Species Mass Fractions, ncyc = 10 t = 5.000000D-071.52707D-01 9.62035D-03 8.37672D-01 Euler1.67403D-01 1.05461D-02 8.22051D-01 midpoint RK
1.70697D-01 1.07537D-02 8.18549D-01 linearized8.28644D-01 analytic
Species Mass Fractions, ncyc = 20 t = 1.000000D-061.56768D-01 9.87614D-03 8.33356D-01 Euler1.56992D-01 9.89031D-03 8.33117D-01 midpoint RK1.57232D-01 9.90538D-03 8.32863D-01 linearized
8.33255D-01 analytic
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Species Mass Fractions, ncyc = 400 t = 2.000000D-051.56789D-01 9.87750D-03 8.33333D-01 Euler1.56789D-01 9.87750D-03 8.33333D-01 midpoint RK1.56789D-01 9.87750D-03 8.33333D-01 linearized
Increasing the time step another factor of 10 produces a poor transient, as shown in the
results on cycle 1. The worst method is Euler, which produces negative mass fractions of O2and H2 on the first cycle. However, all three methods converge to the correct equilibrium.
+++++ Table 5. Increase delt by 1000 (500 ns) +++++
Species Mass Fractions, ncyc = 1 t = 5.000000D-07-8.66921D+00-5.46148D-01 1.02154D+01 Euler
1.09750D+01 6.91408D-01-1.06664D+01 midpoint RK4.92171D-01 3.10061D-02 4.76823D-01 linearized
8.28644D-01 analytic
Species Mass Fractions, ncyc = 2 t = 1.000000D-06-8.66921D+00-5.46148D-01 1.02154D+01 Euler
1.80428D+05 1.13667D+04-1.91794D+05 midpoint RK2.81624D-01 1.77419D-02 7.00634D-01 linearized
8.33255D-01 analytic
Species Mass Fractions, ncyc = 40 t = 2.000000D-05-8.66921D+00-5.46148D-01 1.02154D+01 Euler
--- --- --- midpoint RK1.56789D-01 9.87750D-03 8.33333D-01 linearized
Increasing the time step to 500 ns finally shows failure of both the Euler method and the
Runge-Kutta method. Both produce large negative concentrations on cycle 1, and things get
worse as the calculation proceeds. The Runge-Kutta method fails catastrophically, producing
a NaN on cycle 4. Even though the linearized implicit method is stable, the transient has
substantial truncation errors.
These results are a reminder that a stable solution is not necessarily an accurate
solution. The linearized implicit model has excellent stability properties, but it should
include a time step control that prevents any concentration or the temperature from changing
too much on any one time step.
6.2.2 Case 2
Case 1 was about the simplest case possible, and next we wanted to try something a bit morechallenging. Case 2 differs from Case 1 in having different initial concentrations plus a fourth
inert species (argon, MAr = 39.948). The initial concentrations are C1(0) = 1.60529 105,
C2(0) = 2.40161 105, C3(0) = 9.43882 109, and C4(0) = 1.20554 108. In addition,
this time we included the heats of formation so we could track the evolution of temperature.
The initial temperature was 1500 K. The results for t = 0.5 ns are summarized in Table 6.
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Table 6. Numerical Solution for Case 2
Species Mass Fractions, ncyc = 0 t = 0.000000D+001 2 3 4 T
9.12825D-01 8.60336D-02 2.85269D-04 8.55806D-04 1.50000D+03 t=0.0
Species Mass Fractions, ncyc = 1 t = 5.000000D-10
9.01864D-01 8.53430D-02 1.19371D-02 8.55806D-04 1.48590D+03 Euler9.01974D-01 8.53499D-02 1.18206D-02 8.55806D-04 1.48604D+03 midpoint RK9.02079D-01 8.53566D-02 1.17082D-02 8.55806D-04 1.48617D+03 linearized
1.18215D-02 analytic
Species Mass Fractions, ncyc = 100 t = 5.000000D-083.70361D-01 5.18591D-02 5.76924D-01 8.55806D-04 7.39364D+02 Euler3.72251D-01 5.19782D-02 5.74915D-01 8.55806D-04 7.42286D+02 midpoint RK3.74113D-01 5.20955D-02 5.72936D-01 8.55806D-04 7.45164D+02 linearized
5.74926D-01 analytic
Species Mass Fractions, ncyc = 200 t = 1.000000D-072.05712D-01 4.14864D-02 7.51946D-01 8.55806D-04 4.78566D+02 Euler
2.06935D-01 4.15635D-02 7.50646D-01 8.55806D-04 4.80536D+02 midpoint RK2.08147D-01 4.16398D-02 7.49357D-01 8.55806D-04 4.82488D+02 linearized
7.50652D-01 analytic
Species Mass Fractions, ncyc = 500 t = 2.500000D-078.06996D-02 3.36108D-02 8.84834D-01 8.55806D-04 2.77626D+02 Euler8.09999D-02 3.36297D-02 8.84515D-01 8.55806D-04 2.78104D+02 midpoint RK8.12998D-02 3.36486D-02 8.84196D-01 8.55806D-04 2.78581D+02 linearized
8.84516D-01 analytic
Species Mass Fractions, ncyc = 1000 t = 5.000000D-075.87683D-02 3.22292D-02 9.08147D-01 8.55806D-04 2.42779D+02 Euler5.87965D-02 3.22310D-02 9.08117D-01 8.55806D-04 2.42823D+02 midpoint RK5.88249D-02 3.22328D-02 9.08087D-01 8.55806D-04 2.42868D+02 linearized
9.08117D-01 analytic
All three numerical methods agree quite well and with the analytic solution. All three
seem to be converging to the correct equilibrium OH mass fraction, 0.909598. Note that
this reaction is strongly endothermic and would be quenched if we had used the physical
temperature dependence of the rates instead of the same constant rates used in Case 1.
The behavior of the three methods is the same as for Case 1 as we vary the time step.
Again, the linearized implicit method lets us run with the largest time step. Table 7 is a
short summary of results for t = 5.0 10
7 s.
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Table 7. Numerical Solution for Case 2
Species Mass Fractions, ncyc = 1 t = 5.000000D-073.91721D-01 5.32048D-02 5.54218D-01 8.55806D-04 7.72349D+02 linearized
Species Mass Fractions, ncyc = 1000 t = 5.000000D-045.74031D-02 3.21432D-02 9.09598D-01 8.55806D-04 2.40613D+02
The results for cycle 1 of Table 7 are at the same time as cycle 1000 in Table 6. There is a
big difference between the two solutions. Although accuracy of the transient is poor for the
larger time step, the solution is converging to the correct equilibrium by cycle 1000.
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7 A 21-Species Example
In this section, we consider a more complex kinetics mechanism. It uses 21 species and
30 reactions. Ar, H2O2, C2H2, and CH2O are treated as inert. No hydrocarbon fuels are
included in the initial conditions so the chemistry is mainly that of oxygen and hydrogen,
minus any H2O2 chemistry. The initial conditions are P = 1.013 106
dynes/cm2
, =1.25029104 g/cm3, and T = 1700 K. Table 8 lists the molecular weights, heats of formation,
and equilibrium constants for the 21 species. Table 9 is a list of the forward rate constants
for the 30 reactions.
Table 10 is a summary of computational results. It shows temperatures and species
mass fractions at 10 s for two numerical methods and three values of t (2.5, 5.0, and
20.0 ns). The second-order Runge-Kutta method is denoted by midpoint RK, and the
first-order linearized implicit method is denoted by linearized.
Consider first the inert species Ar, H2O2, C2H2, and CH2O. Only Ar is inert in the
sense of not normally being chemically reactive. However, the other 3 species are not included
in any of the chemical reactions in this mechanism (except all species are included in the
chemical symbol M). Therefore the mass fractions of this set of species should be constant
in time. This is true for all four inert species for both numerical methods and for all three
time steps.
Next consider the reactive hydrocarbon species CH4, C2H4, and C3H8. Both the
initial mass fractions and the reverse rates of their global oxidation reactions were set to
zero. The mass fractions remained zero for the Runge-Kutta integration for all three time
steps. For the linearized method, the mass fractions are all tiny negative numbers. Theseare due to rounding errors in the linear-system solver. 4 The worst case is a mass fraction of
1.0 1024 for methane. This represents roughly one methane molecule in a mole of fluid,
which is physically insignificant. These small negative numbers are merely a minor cosmetic
issue, as long as they stay small.
For the 0.25 ns time step, the two numerical methods agree to about one part in 104
for all species except N, where the difference is about 1 percent. This is a trace species, so
the agreement between the two methods suggests that this solution is time-step-independent
for all practical purposes.
For the Runge-Kutta method, doubling the time step to 5.0 ns makes a little differ-
ence, and the agreement is still mostly at the 1 part in 104 level. The exceptions are N and
CHO. The N mass fraction is still fairly accurate, but that of CHO now is in error by about
30 percent. For the first-order linearized implicit method, the worst discrepancy between
4All calculations were done in double precision (real*8).
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the two time steps is about one percent in the mass fractions. This includes N, and there is
almost no change in CHO. Note also that the predicted temperatures for the four solutions
discussed so far agree to within 5 K.
Next the time step was increased by a factor of 4 to 20.0 ns. The Runge-Kutta
solution is now totally unacceptable. The mass fraction of OH is -0.1. OH is a key radical
for the combustion of all hydrocarbons, and such a large negative mass fraction renders
the solution physically meaningless. On the other hand, the linearized implicit method
is still well-behaved. The only negative mass fractions are the tiny rounding errors for
methane, ethylene, and propane. With few exceptions, the truncation errors are on the
order of 3 percent. The mass fraction of N has decreased almost 10 percent for N with the
Runge-Kutta method, but has increased by almost the amount with the first-order linearized
implicit method. N2O and CHO are significantly more accurate for the first order method.
The second-order mass fraction of CHO is off by 3 orders of magnitude (possibly influenced
by the massive error in OH). The final temperatures increased by 45 K for the Runge-Kuttasolution, but only by 18 K for the first-order method.
In closing, we make the following observation: The formal order of accuracy for
a numerical method says little about its accuracy. Even for the largest time step that
produced a large negative mass fraction of OH, the Runge-Kutta method kept on running.
It was not unstable in the sense of the solution becoming unbounded, but the solution is
physically unrealistic. The first-order method seems to be less prone to producing negative
mass fractions, and its accuracy at large time steps for species such as CHO may be related
to its ability to keep fast reactions in near-equilibrium under conditions that would cause
stability problems with explicit methods.
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Table 8. Species Thermodynamic Parameters
Species M H A b E
1 CH4 1.604296D+01 -1.59921128D+01 1.51456D-06 0.00000D+00 -2.19712D+042 O2 3.199880D+01 0.00000000D+00 1.00000D+00 0.00000D+00 0.00000D+003 N2 2.801348D+01 0.00000000D+00 1.00000D+00 0.00000D+00 0.00000D+004 CO2 4.401000D+01 -9.39653441D+01 1.12780D+01 -2.86036D-01 -9.37359D+045 H2O 1.801528D+01 -5.71034894D+01 2.03999D-02 -3.56445D-01 -5.86397D+046 H 1.007940D+00 5.16336042D+01 1.61801D+01 5.14866D-01 5.21438D+04
7 H2 2.015880D+00 0.00000000D+00 1.00000D+00 0.00000D+00 0.00000D+008 O 1.599940D+01 5.89842256D+01 2.64100D+02 2.86036D-01 5.98602D+049 N 1.400674D+01 1.12528680D+02 1.27475D+02 3.69647D-01 1.13213D+05
10 OH 1.700734D+01 9.17543021D+00 2.46399D+01 -1.80423D-01 9.48250D+0311 CO 2.801060D+01 -2.72000478D+01 4.27854D+07 -8.75711D-01 -2.49755D+0412 NO 3.000614D+01 2.14567399D+01 4.60300D+00 0.00000D+00 2.16200D+0413 Ar 3.994800D+01 0.00000000D+00 1.00000D+00 0.00000D+00 0.00000D+0014 C2H4 2.805416D+01 1.45760038D+01 3.81096D-04 -2.50832D-01 9.41822D+0315 C3H8 4.409712D+01 -1.94400000D+01 8.86700D-15 -5.90400D-01 -2.76300D+0416 HO2 3.300674D+01 1.19646271D+00 3.74583D-03 -3.08039D-02 -3.35439D+02
17 H2O2 3.401468D+01 -3.10248565D+01 2.07494D-05 -3.34443D-01 -3.28504D+0418 N2O 4.401288D+01 2.04304493D+01 1.44700D+00 -3.75100D-01 3.42000D+0419 C2H2 2.603828D+01 5.63467973D+01 5.35000D+02 0.00000D+00 5.30100D+0420 CH2O 3.002648D+01 -2.67822657D+01 9.05000D-03 0.00000D+00 -3.03700D+0421 HCO 2.901854D+01 1.03133365D+01 1.41000D+02 0.00000D+00 8.67800D+03
The heats of formation H are in units of kcal/mol, and E is in cal/mol (R = 1.9872).
The species equilibrium constants are KP = ATb exp(E/RT) atm (use R = 82.06 atm-
cm3/(K mol) to convert to KC). 4.184 J/cal.
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Table 9. Chemical Mechanism
No. Reaction Aj bj Ej/R (K) Ref.1. O2 + H O + OH 2.00 1014 0.0 8.4551 103 [15]2. H2 + O H + OH 1.80 1010 1.0 4.4416 103 [15]3. H2 + OH H2O + H 1.17 109 1.3 1.8245 103 [15]4. 2 OH H2O + O 6.04 108 1.3 2.766 101 [15]5. H + HO2 2 OH 1.5 10
14 0.0 5.0514 102 [15]6. H + OH + M H2O + M 2.20 1022 -2.0 0.0 [15]7. 2 H + M H2 + M 1.80 1018 -1.0 0.0 [15]8. 2 O + M O2 + M 6.17 1015 -0.5 0.0 [15]9. H + O + M OH + M 6.20 1016 -0.6 0.0 [15]10. H + O2 + M HO2 + M 2.30 1018 -0.8 0.0 [15]11. CO + OH CO2 + H 1.5 107 1.3 3.855 102 [16]12. CO + O2 CO2 + O 2.53 10
12 0.0 2.40036 104 [16]13.1 CO + O + M CO2 + M 1.35 1024 -2.79 2.1085 103 [16]
14.2
C2H4 + O2 2 CO + 2 H2 1.6 1013
0.0 1.5 104
15.3 N2 + M 2 N + M 3.70 1021 -1.6 1.132 105 [17]16. O + N2 NO + N 1.80 1014 0.0 3.8368 104 [18]17. O2 + N NO + O 6.40 10
9 1.0 3.1512 103 [18]18. OH + N NO + H 3.0 1013 0.0 0.0 [18]19.4 N2O + M N2 + O + M 9.13 1014 0.0 2.9036 104 [18]20. N2O + O 2 NO 1.0 10
14 0.0 1.409 104 [16]21. N2O + O N2 + O2 1.0 1014 0.0 1.409 104 [16]22. N2O + H N2 + OH 2.53 1010 0.0 2.2897 103 [16]23. N2O + H N2 + OH (pt. 2) 2.23 1014 0.0 8.4541 103 [16]24.5 CH4 + O2 CO + 2 H2 + O 3.00 1013 0.0 1.500 104
25.6 C3H8 + O2 CH4 + 2 CO + 2 H2 2.30 1013 0.0 1.5000 10426. CO + H2 CHO + H 1.30 1015 0.0 4.52811 104
27. CO + H2O CHO + OH 2.80 1015 0.0 5.28186 104
28. CO + HO2 CHO + O2 6.70 1012 0.0 1.62451 104
29. CO + H + M CHO + M 1.14 1015 0.0 1.20031 103
30. N2O + OH N2 + HO2 2.00 1012 0.0 1.05978 104
kf = AjTbj exp(Ej/RT), units are cm, mol, kJ, and K (R = 8.31451 103).
1Third body efficiencies: CO2 = 3.8, H2O = 12.0, H2 = 2.5, CO = 1.9.2 [C2H4]
0.75[O2]
3Third body efficiency: N = 4.5.4Third body efficiencies: Ar = 0.63, H2O = 7.5.5 [CH4]
0.75[O2]6 [C3H8]
0.75[O2]
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8 Summary and Conclusions
We investigated a first-order method for integrating the rate equations for chemical kinetics.
It is based on an implicit method that is linear in advanced-time quantities. It is compu-
tationally less work than the fully implicit method. The amount of computational effort is
approximately the same as for one iteration of the fully implicit method when solved by thestandard Newton-Raphson method. When compared with the Euler method and a second-
order Runge-Kutta method base on the mid-point rule, the linearized-implicit method has
improved accuracy and stability properties at larger time steps.
Work is in progress to see if this translates into being able to simulate combustion in
simple burners more quickly and with more accuracy than is possible with the explicit meth-
ods. At the time of this writing, a detailed two-dimensional simulation of a laminar Bunsen
burner flame is in progress that couples the full fluid dynamics equations with the skeletal
mechanism for methane combustion and NOx production given in Table 1 of Glarborg, et
al. [19]. This mechanism has 27 species and 77 reactions. I supplemented the reaction set
with 9 additional reactions to aid in thermal ignition. So far the linearized implicit method
is performing well. The midpoint Runge-Kutta method was numerically unstable for t of
20 ns, but the linearized implicit method is stable at 200 ns.
In addition, a one-dimensional laminar H2-air flame has been simulated with the GRI
3.0 mechanism for methane combustion [20], which includes 53 species and 325 reactions.
There is excellent agreement between the linearized implicit method and Runge-Kutta when
run with a time step for which the Runge-Kutta method performed well. Further numerical
experimentation will be needed to find the limits of accuracy and stability in this method.A task for future code development will be to extend this linearly implicit method
to second order by using the linearly implicit algorithm in both the predictor and corrector
steps of the midpoint-rule Runge-Kutta method.
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References
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Ap. J. Suppl. 124, 241 (1999).
[2] L. D. Cloutman, 1990, COYOTE: A Computer Program for 2D Reactive Flow Simu-
lations, Lawrence Livermore National Laboratory report UCRL-ID-103611, 1990.
[3] D. G. Cloutman and L. D. Cloutman, A Unified Mathematical Framework for Popu-
lation Dynamics Modelling, Ecol. Modelling 71, 131 (1993).
[4] L. H. Aller and S. Chapman, Diffusion in the sun, Ap. J. 132, 461 (1960).
[5] R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport Phenomena (Wiley, New
York, 1960).
[6] S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases
(Cambridge University Press, London, 1952).
[7] J. D. Ramshaw, Self-consistent effective binary diffusion in multicomponent gas mix-
tures, J. Non-Equilib. Thermodyn. 15, 295 (1990).
[8] J. D. Ramshaw, Hydrodynamic theory of multicomponent diffusion and thermal dif-
fusion in multitemperature gas mixtures, J. Non-Equilib. Thermodyn. 18, 121 (1993).
[9] J. D. Ramshaw and C. H. Chang, Ambipolar Diffusion in Multicomponent Plasmas,
Plasma Chem. Plasma Proc. 11, 395 (1991).
[10] J.-L. Tassoul, Theory of Rotating Stars, (Princeton U. Press, Princeton, 1978).
[11] D. D. Clayton, Principles of Stellar Evolution and Nucleosynthesis (McGraw-Hill, New
York, 1968).
[12] G. R. Caughlin and W. A. Fowler, Thermonuclear reaction rates V, At. Data Nucl.
Data Tables 40, 284 (1988).
[13] J. D. Ramshaw, Elements of Computational Fluid Dynamics (Imperial CollegePress/World Scientific, London/Singapore, 2011).
[14] C. K. Westbrook and F. L. Dryer, Simplified reaction mechanisms for the oxidation of
hydrocarbon fuels in flames, Combustion Sci. Tech. 27, 31 (1981).
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[15] C. J. Montgomery, G. Kosaly, and J. J. Riley, Direct numerical simulation of turbulent
reacting flow using a reduced hydrogen-oxygen mechanism, Comb. Flame 95, 247
(1993).
[16] M. T. Allen, R. A. Yetter, and F. L. Dryer, High pressure studies of moist carbon
monoxide/nitrous oxide kinetics, Combust. Flame 109, 449 (1997).
[17] C. Park, Assessment of a two-temperature kinetic model for dissociating and weakly
ionizing nitrogen, J. Thermophysics 2, 8 (1988).
[18] J. Warnatz, Concentration-, pressure-, and temperature-dependence of the flame ve-
locity in hydrogen-oxygen-nitrogen mixtures, Combust. Sci. Technol. 26, 203 (1981).
[19] P. Glarborg, N. I. Lilleheie, S. Byggstoyl, and B. F. Magnussen, A Reduced Mech-
anism for Nitrogen Chemistry in Methane Combustion, Twenty-Fourth Symposium
(International) on Combustion, The Combustion Institute, 1992, pp. 889-898.
[20] G. P. Smith, D. M. Golden, M. Frenklach, N. W. Moriarty, B. Eiteneer, M. Goldenberg,
C. T. Bowman, R. K. Hanson, S. Song, W. C. Gardiner, Jr., V. V. Lissianski, and Z.
Qin. http:www.me.berkeley.edugri mech
[21] L. D. Cloutman, A Note on the Stability and Accuracy of Finite Difference Approx-
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UCRL-ID-125549, 1996.
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A Fully Implicit Chemistry Algorithm
Timmes [1] outlines three implicit methods for integrating nuclear reaction networks. The
simplest one is equivalent to the method outlined in Section 5, although he describes it in
slightly different language. Following Timmes, we consider the system of equations
y = f(y). (75)
Define the Jacobian
J =f
y. (76)
Let a superscript n denote the solution at time step n. Then we assume
yn+1 = yn + t f(yn+1) yn + tJ
yn+1 yn
, (77)
where t is the time step. If we let = yn+1 yn, then equation (77) may be rearranged to
(U t J) = t f(yn), (78)
where U is the unit matrix. This linear system may be solved for using any standard
linear system solver, and then yn+1 may be computed.
A fully implicit method may be similarly derived. Define a function g(y()) such that
g(yn+1) = yn+1 yn t f(yn+1) = 0. (79)
If we have an estimate ofyn+1
, sayy()
, then
g(y(+1)) = g(y()) + G(+1)
y(+1) y()
= 0. (80)
The last equality provides the basis for an iterative solution method for yn+1. Unfortunately,
evaluation ofG(+1) = g/y(+1) is as expensive as evaluating J. However, it is likely that
the iterations will converge ifG(+1) is held fixed after being initialized on the first iteration.
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B The Logistic Equation: A Warning
The logistic equation is the basis of a crude but popular and useful model of population
dynamics [3]. If N is the population, then the hypothesis is that it obeys the logistic
equationdN
dt = ALN BLN2 = ALNKL N
KL
, (81)
where KL = AL/BL is the carrying capacity of the environment. The first term on the right
hand side represents the familiar exponential growth of the population via constant difference
between birth and mortality rates. The nonlinear term is a simple ad hoc representation of
an increase in mortality or decrease in fertility due to crowding.
The logistic equation has a very simple general solution. At first glance this equation
appears to have two free parameters, but it can be reduced to a dimensionless equation that
has a self-similar solution with no parameters. Let us make the linear changes of variables
N = N/KL and = ALt. Then equation (81) becomes
dN
d= N N2. (82)
We assume that the value of N(0) is given. Equation (82) is easily solved by separation of
variables:
N() =N(0)
N(0) + [1 N(0)] exp(). (83)
There are two equilibrium solutions, N = 0 and 1.
The equilibrium solution N() = 1 is stable. If the solution is perturbed such that
N() > 1, the solution decays monotonically to unity. If the equilibrium is perturbed by
decreasing the population, N grows monotonically back to unity.
The equilibrium solution N = 0 is unstable. For a positive perturbation, the solution
follows equation (83) to unity as increases. For a negative perturbation, the formal solution
diverges to in a finite time. A negative perturbation is not realizable physically.
The ecologically interesting case is 0 < N(0) < 1. In this case, the solution rises
monotonically to unity at late times. The solution does not overshoot the carrying capacity.
The nature of the solution is independent of the parameters A and B. These parameters
determine the carrying capacity and the time scale for population growth, but not the shapeof the curve of N versus t.
We note that populations obeying equation (81) do not exhibit chaos. Ordinary
differential equations that are local in the independent variable and have no external forcing
terms must be of at least third order to have chaotic solutions.
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7. For A greater than 4, the solution diverges to .
When using the approximation equation (84) to solve numerically equation (82), the
first possibility, A < 1, is not physically meaningful. The second is the range of A that
produces meaningful approximations to equation (83), although accuracy decreases as A
increases. Possibilities 3 through 6 represent bounded solutions to equation (84) that arequalitatively different than the solution to equation (82). The final possibility represents a
traditional numerical instability.
Other ways of differencing the logistic equation show similar types of behavior, al-
though the details can differ considerably. Each particular difference approximation has its
own behavior, but typically there are a region of stability for small time steps, then oscilla-
tory and periodic solutions, chaotic solutions, and eventually solutions that become infinite.
In particular, it is noteworthy that the two second-order Runge-Kutta methods discussed in
previous sections can converge to the wrong steady state value for a limited range of time
steps [21].
So what does all this have to do with combustion simulations? Simple: the finite
difference equations for combustion simulations, with or without coupling with fluid dynam-
ics, are simply very complicated nonlinear iterated maps with t as a free parameter. The
present result suggests that we should not be surprised to see the same progression of solution
types in combustion simulations as in the logistic map: smooth solutions, various oscillatory
solutions, chaotic solutions, and finally instability as t is increased. This behavior has in
fact been observed [21].